Two entrepreneurs, A and B, working out of a single
Question # 00549685
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Updated on: 06/21/2017 01:24 AM Due on: 06/21/2017
1. Two entrepreneurs, A and B, working out of a single enterprise are considering the development of a new app. Each one of them has to decide, simultaneously but independently, whether to invest in creating it. If an entrepreneur decides to invest, his cost is $30,000. They both derive revenue from the app once offered for sale since they are business partners. If only one of them invests in the app, they each get a revenue of $20,000 (regardless of which one of them incurred the investment cost of $30,000). If both of them invest, they can make the app more advanced and can charge higher prices. They will each get a revenue of $40,000 then. If they both decide not to invest, they each receive 0 revenue (and incur zero cost).
(a) Draw the payoff matrix showing the profit (revenue - development cost) for each entrepreneur.
(b) What is A's best strategy? What is best B's strategy? Does any one of them have a dominant strategy? Based on your previous answer, what will be the outcome of the game?
(c) Are A and B in a prisoner's dilemma?
2. MODIFY Example 4.2 in Game Theory 1: Dominant Strategies by making just one change: when the total output is 6, the price is $22.(The price $50 of total output is 4, and $20 when it is 8). Derive the new 2x2 matrix depicting the modified game between Kuwait and Iran. Does Kuwait have a dominant strategy? Does Iran have a dominant strategy? Can you advise Kuwait on how much it should produce?
Introduction
The main aim of this section of Module 4 is to formally define the basic concepts of game theory, including dominant
strategies and Nash equilibrium, and illustrate them through some simple examples. Much of this should already be
familiar to you from Module 2; see Lecture Materials 2.3 (https://canvas.brown.edu/courses/1070421/pages/lecturematerials-2-dot-3-a-primer-in-game-theory-through-classroom-experiments-de-clippel) . The main supplemental reading is Chapter 3, The Art of Strategy. Please read that as we go along. Decision Theory
Suppose you have to choose one object from a set of alternatives,
get from any particular choice, , i.e., there is a function , where . Suppose there is a way to measure the utility you
denotes the utility from choosing . What does ) that maximizes your utility, rationality mean? It means that you make a choice (from . Rational Choice
A rational/optimal/utility-maximizing choice is , where for all in . The solution to this constrained maximization problem describes rational behavior. Economics is a social science, concerned ultimately with social rather
than individual outcomes. Thus, it is important to study rational individual behavior in a world with many individuals
interacting with each other. Game Theory
Game theory refers to interactive decision theory. There are two or more decision makers (players), each making a rational
choice for herself. Each player’s utility depends on her own choice as well as choices of others.
"Game theory, except in trivial cases, propounds paradoxes rather than solves problems"
- Paul Samuelson,1969
"To know game theory is to change your lifetime way of thinking"
- Paul Samuelson, 2004 Two Player Games
The players are called A and B. Player A has two strategies, called “Up” and “Down”. Player B has two strategies, called
“Left” and “Right”. The table below shows the payoffs to both players for each of the four possible strategy combinations is
the game’s payoff matrix.
Player B
L Player A R U ( a, b ) ( c, d ) D ( e, f ) ( g, h ) https://canvas.brown.edu/courses/1070421/pages/game-theory-i-dominant-strategies?modu... 6/17/2017 Game Theory I: Dominant Strategies: EMSL2200: Economic Perspectives on Strategic D... Page 2 of 4 Shown above is the game’s payoff matrix. Player A’s payoff is shown first. Player B’s payoff is shown second. Dominant Strategies
Strategy U is a dominant strategy for player A if it is better than D regardless of whether B chooses L or R: If U is A’s dominant strategy, rationality demands that A plays U. Suppose each player has a dominant strategy. Then we
can conclude that if both players are rational, these are the strategies they will chooses to play. A pair of dominant
strategies is also known as a dominant strategy equilibrium; it describes how the game will be played.
We now turn to a well-known example of a game in which dominant strategies, despite the rationality they embody, may
not lead to what is best for the two players. Example 4.1 is not exactly the same as the example of a prisoner's dilemma
you saw in Module 2, but essential idea is the same, and it is worth repeating. EXAMPLE 4.1: The Prisoner's Dilemma
Two prisoners (Bonnie and Clyde) are being interrogated, separately. The DA has limited evidence. Each prisoner can
either confess or remain silent. The length of each prisoner’s jail term depends on what both of them decide.
If both remain silent: each one gets 5 years in jail:
Clyde
Silent
Bonnie Silent Confess ( -5, -5 ) Confess If Bonnie is silent, but Clyde confesses: 30 years in jail for Bonnie, only 1 for Clyde
Clyde
Silent
Bonnie Silent Confess
( -30, 1 ) Confess If Bonnie confesses but Clyde is silent: 1 year in jail for Bonnie, 30 years in jail for Clyde.
Clyde
Silent
Bonnie Silent Confess https://canvas.brown.edu/courses/1070421/pages/game-theory-i-dominant-strategies?modu... 6/17/2017 Game Theory I: Dominant Strategies: EMSL2200: Economic Perspectives on Strategic D... Page 3 of 4 ( -1, -30 ) Confess If both Bonnie and Clyde confess, they each get 10 years.
Clyde
Silent
Bonnie Confess Silent
( -10, -10 ) Confess Dominant Strategies in The Prisoner's Dilemma
If Bonnie plays Confess then Clyde’s best reply is Confess. If Bonnie plays Silence then Clyde’s best reply is Confess. No
matter what Bonnie plays, Clyde’s best reply is always Confess. Confess is a dominant strategy for Clyde.
Clyde Silent Silent Confess ( -5, -5 ) ( -30, -1 ) Bonnie
Confess ( -1, -30) ( -10, -10 ) Clyde
Silent Confess Silent ( -5, -5 ) Confess ( -1, -30) ( -10, -10 ) ( -30, -1 ) Bonnie Similarly, no matter what Clyde plays, Bonnie’s best reply is always Confess. Confess is a dominant strategy for Bonnie
also.
Clyde Bonnie Silent
Confess Silent Confess ( -5, -5 ) ( -30, -1 ) ( -1, -30) ( -10, -10 ) https://canvas.brown.edu/courses/1070421/pages/game-theory-i-dominant-strategies?modu... 6/17/2017 Game Theory I: Dominant Strategies: EMSL2200: Economic Perspectives on Strategic D... Page 4 of 4 Clyde
Silent
Bonnie Confess Silent ( -5, -5 ) Confess ( -1, -30) ( -10, -10 ) ( -30, -1 ) The only dominant strategy equilibrium is (C,C), even though (S,S) gives both Bonnie and Clyde better payoffs. The
equilibrium is not socially rational! What does this have to do with economics?
An industry with a small number of competing firms is known as an oligopoly. In such an industry, the profit of a firm depends
crucially on not only its own output decision (its strategy) but also on the decisions/strategies of its rivals. And the firms could
face a situation that is like a prisoner’s dilemma. We'll be studying this in more depth in Module 5, but for now a simple
example will suffice to show how the prisoner's dilemma may be relevant for two competing firms. Example 4.2: OPEC
Kuwait and Iran can each produce just two production levels: either 2 or 4 million barrels of crude oil a day. The total output is
therefore 4, 6 or 8. The price will be $50, $30, $20 per barrel, respectively. Cost of production is 0. How much will each
produce? This information can easily be translated into a 2x2 game.
If prices are $50, $30, $20:
Kuwait
2
Iran 4 2 (100, 100) (60, 120) 4 (120, 60) (80, 80) It is easy to see that playing 4 is dominant strategy for each player. In the absence of cooperation, therefore, they earn 80
each. If they collude (and agree to produce only 2 each) they can each earn 100 > 80. They are in a prisoner’s dilemma. More General Games
The prisoner’s dilemma is an example of how individual rationality may not imply social rationality. This message is of limited
value because not all games are such that players have dominant strategies. To understand more general games we will
need to introduce another equilibrium concept: Nash Equilibrium. This will allow us to provide a more thorough analysis of an
oligopoly. https://canvas.brown.edu/courses/1070421/pages/game-theory-i-dominant-strategies?modu... 6/17/2017
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Solution: Two entrepreneurs, A and B, working out of a single